Topic Review — Continuous Random Variables and Normal Distribution
← Continuous RV and Normal Distribution
This review covers all lessons in this topic: continuous random variables and PDFs, expected value and variance of continuous RVs, the normal distribution, and using the normal distribution (inverse normal, finding parameters).
Review Questions
- State two conditions that a probability density function (PDF) f(x) must satisfy.
- A continuous random variable X has PDF f(x) = kx for 0 ≤ x ≤ 4 and 0 elsewhere. Find k.
- For the PDF f(x) = 1/8 · x on [0, 4], find P(1 < X < 3).
- A continuous random variable has PDF f(x) = 3x² for 0 ≤ x ≤ 1. Find E(X).
- For the PDF f(x) = 3x² on [0,1], find E(X²) and hence Var(X).
- X ~ N(80, 25). Find: (a) μ and σ (b) P(X < 85) using CAS (c) P(70 < X < 90)
- Use the 68–95–99.7 rule to answer: X ~ N(200, 400) (so σ = 20). What percentage of values are greater than 240?
- The time (in minutes) students spend on homework daily is normally distributed with mean 45 min and standard deviation 10 min. Find the probability that a randomly selected student spends less than 30 minutes on homework.
- X ~ N(60, 16). Find the value c such that P(X > c) = 0.1.
- X ~ N(μ, 49). Given P(X < 30) = 0.6, find μ.
- X ~ N(50, σ²). Given P(X < 60) = 0.8, find σ.
- Weights of apples are normally distributed. P(X < 150) = 0.2 and P(X < 200) = 0.9. Find the mean and standard deviation.
- A normal distribution has the property that P(X < 42) = P(X > 58). What is the mean? Explain your reasoning.
- A continuous random variable X has PDF f(x) = c(4 − x²) for −2 ≤ x ≤ 2 and 0 elsewhere.
- (a) Find c
- (b) Find E(X) and explain the result using symmetry
- (c) Find P(0 < X < 1)
- Scores on a standardised test are normally distributed with μ = 500 and σ = 100. A scholarship requires a score in the top 5%.
- (a) Find the minimum score required for the scholarship.
- (b) What percentage of students score between 400 and 650?
- (c) If 2000 students sit the test, how many are expected to score above 700?